We can see that 28 is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

My Algorithm

Similar to other problems, my solution consists of two steps
1. precompute all possible inputs
2. for each test case: perform a simple lookup

It takes less than a second to find all such numbers with at most 1000 divisors.
Two "tricks" are responsible to achieve that speed:
You can get all divisors of x by analyzing all potential divisors i<=sqrt{x} instead of i<x.
Whenever we find a valid divisor i then another divisor j=frac{x}{y} exists.
The only exception is i=sqrt{x} because then j=i.

Somehow more subtle is my observation that when numbers have more than about 300 divisors,
the smallest one always end with a zero. I cannot prove that, I just saw it while debugging my code.

I decided to store all my results in a std::vector called smallest wheresmallest[x] contains the smallest triangle number with at least x divisors.

While filling that container, the program encounters many "gaps":
e.g. 10 is the smallest number with 4 divisors and 28 is the smallest number with 6 divisors
but there is no number between 10 and 28 with 5 divisors.
Therefore 28 is the smallest number with at least 5 divisors, too.

Alternative Approaches

Prime factorization can find the result probably a bit faster.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent toecho "1 7" | ./12

Output:

(please click 'Go !')

Note: the original problem's input 500cannot be enteredbecause just copying results is a soft skill reserved for idiots.

(this interactive test is still under development, computations will be aborted after one second)

My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include<iostream>

#include<vector>

intmain()

{

// find the smallest number with at least 1000 divisors

// (due to Hackerrank's input range)

constunsignedint MaxDivisors = 1000;

// store [divisors] => [smallest number]

std::vector<unsignedint> smallest;

smallest.push_back(0); // 0 => no divisors

// for index=1 we have triangle=1

// for index=2 we have triangle=3

// for index=3 we have triangle=6

// ...

// for index=7 we have triangle=28

// ...

unsignedint index = 0;

unsignedint triangle = 0; // same as index*(index+1)/2

while (smallest.size() < MaxDivisors)

{

// next triangle number

index++;

triangle += index;

// performance tweak (5x faster):

// I observed that the "best" numbers with more than 300 divisors end with a zero

// that's something I cannot prove right now, I just "saw" that debugging my code

Changelog

Hackerrank

Difficulty

5%
Project Euler ranks this problem at 5% (out of 100%).

Hackerrank describes this problem as easy.

Note:Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is rarely an option.

Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.

Heatmap

Please click on a problem's number to open my solution to that problem:

green

solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too

yellow

solutions score less than 100% at Hackerrank (but still solve the original problem easily)

gray

problems are already solved but I haven't published my solution yet

blue

solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much

orange

problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte

red

problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too

black

problems are solved but access to the solution is blocked for a few days until the next problem is published

[new]

the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.

The 310 solved problems (that's level 12) had an average difficulty of 32.6&percnt; at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of &approx;60000 in August 2017)
at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.All of my solutions can be used for any purpose and I am in no way liable for any damages caused.You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.Thanks for all their endless effort !!!

more about me can be found on my homepage,
especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !